Homogenization of Parabolic Equations with an Arbitrary Number of Scales in Both Space and Time. (24th February 2014)

• Record Type: Journal Article
• Authors: Flodén, Liselott; Holmbom, Anders; Olsson Lindberg, Marianne; Persson, Jens
• Other Names: Conca Carlos Academic Editor.
• Is Part Of: Journal of applied mathematics. Volume 2014(2014)
• Publication Date: 2014-02-24
• Subjects: Mathematics -- Periodicals; 519.05
• DOI: 10.1155/2014/101685
• Languages: English
• ISSNs: 1110-757X

Abstract

Abstract : The main contribution of this paper is the homogenization of the linear parabolic equation ∂ t u ε ( x, t ) - ∇ · ( a ( x / ε q 1, . . ., x / ε q n, t / ε r 1, . . ., t / ε r m ) ∇ u ε ( x, t ) ) = f ( x, t ) exhibiting an arbitrary finite number of both spatial and temporal scales. We briefly recall some fundamentals of multiscale convergence and provide a characterization of multiscale limits for gradients, in an evolution setting adapted to a quite general class of well-separated scales, which we name by jointly well-separated scales (see appendix for the proof). We proceed with a weaker version of this concept called very weak multiscale convergence. We prove a compactness result with respect to this latter type for jointly well-separated scales. This is a key result for performing the homogenization of parabolic problems combining rapid spatial and temporal oscillations such as the problem above. Applying this compactness result together with a characterization of multiscale limits of sequences of gradients we carry out the homogenization procedure, where we together with the homogenized problem obtain n local problems, that is, one for each spatial microscale. To illustrate the use of the obtained result, we apply it to a case with three spatial and three temporal scales with q 1 = 1, q 2 = 2, and 0 < r 1 < r 2 .